A generalization of a theorem of Jacobson

Abstract:

A well-known theorem of Jacobson asserts that a ring $R$ in which ${x^{n(x)}} = x$ for each $x$ in $R$ must be commutative. This paper gives a description of a ring with involution in which the above condition is imposed only on the symmetric elements. In particular, if $R$ is primitive, $R$ is either commutative or the $2 \times 2$ matrices over a field, and, in general, any such $R$ is locally finite and satisfies a polynomial identity of degree 8.